The first law for the layered cube [frap-0004]
✍️source

    The layered model needs a reachability test. A tuple of permutations and orientations can describe a dismantled cube without being obtainable by legal slice turns. We now choose the labelled, fixed-frame state space from the layered-cube model and state the first law for that space. The theorem is a statement about labelled cubies; an ordinary colour pattern is its observation after equal-looking labels have been forgotten.

    1. Coordinates for one state space

      Let \(K_o=(N-3)/2\) for odd \(N\) and \(K_e=N/2-1\) for even \(N\). For every cube there are a corner permutation \(\sigma \in S_8\) and corner orientation vector \(x\in (\mathbb Z_3)^8\). For odd \(N\), there is also a single-edge permutation \(\tau _s\in S_{12}\) and flip vector \(z\in (\mathbb Z_2)^{12}\). Each coupled-edge orbit indexed by \(i\in I\) has a permutation \(\tau _i\in S_{24}\) and a labelled orientation vector \(y_i\in (\mathbb Z_2)^{24}\). Each center-corner orbit has a permutation \(\rho _i\in S_{24}\). Center-edge orbits are indexed by a finite set \(J\); write \(\rho _{ij}\) for the orbit whose two distance signatures are \(i,j\). In the fixed-center convention, \(I=\{1,\ldots ,K_o\}\) and \(J=I^2\) when \(N\) is odd, while \(I=\{1,\ldots ,K_e\}\) and \(J=\{(i,j)\in I^2:i\ne j\}\) when \(N\) is even. This indexing records which coupled-edge signatures border a center-edge orbit; it is the coordinate version of the rings in the preceding chapter.

      The full labelled configuration space is the product of these factors, with the solved state represented by identity permutations and zero orientation vectors. It is deliberately larger than the move group. For example, a vector \(y_i\) records all sticker orientations allowed by a labelled reassembly, even though the legal slice action fixes the relevant coupled-edge orientation in its orbit. Whole-cube rotations are still outside the generator set, and the six face directions remain part of the frame.

      2. The invariant law

        For odd \(N\), a configuration is legal exactly when

        \[\begin {aligned} \operatorname {sgn}(\sigma )&=\operatorname {sgn}(\tau _s)= \operatorname {sgn}(\rho _i) &&(i\in I),\\ \operatorname {sgn}(\rho _{ij})&=\operatorname {sgn}(\sigma ) \operatorname {sgn}(\tau _i)\operatorname {sgn}(\tau _j) &&((i,j)\in J),\\ \sum _r x_r&=0\pmod 3,\qquad \sum _r z_r=0\pmod 2,\\ y_i&=0 &&(i\in I). \end {aligned}\]

        For even \(N\), the corresponding law is

        \[\begin {aligned} \operatorname {sgn}(\sigma )&=\operatorname {sgn}(\rho _i) &&(i\in I),\\ \operatorname {sgn}(\rho _{ij})&=\operatorname {sgn}(\sigma ) \operatorname {sgn}(\tau _i)\operatorname {sgn}(\tau _j) &&((i,j)\in J),\\ \sum _r x_r&=0\pmod 3,\\ y_i&=0 &&(i\in I). \end {aligned}\]

        The empty-index cases are part of the statement: \(N=2\) has only corners, \(N=3\) has the usual twelve single edges and fixed centers, and \(N=4\) has one coupled-edge and one center-corner orbit but no center-edge orbit. The \(N=5\) instance has one orbit of each coupled-edge, center-corner and center-edge type, giving the 98 visible cubies counted earlier.

        Necessity follows by checking generators. An outer face turn is a 4-cycle on corners, single edges, center corners and each relevant center-edge orbit, and two 4-cycles on each affected coupled-edge orbit. An inner slice turn is the identity on corners and single edges, and is a 4-cycle on its coupled edges and on the matching center-edge orbits, with two 4-cycles on center corners. These cycle signs give the two parity rows. The corner character changes by two opposite units on a non-\(U\)/\(D\) face and not at all on a \(U\)/\(D\) face, so its sum is fixed modulo 3. An outer turn flips four single edges, so their sum is fixed modulo 2. The chosen coupled-edge sticker convention makes every legal \(y_i\) zero. The solved state satisfies the rows, hence every word in the generators does.

        Bonzio--Loi--Peruzzi give the labelled Professor's-Cube version of the same tuple and its corrected coupled-edge condition [bonzio2018nxnxn, Theorem 5]. Salkinder states the odd and even families and checks these generator effects in [salkinder2021nxnxn, Theorems 8--9]. Their notation indexes center-edge orbits by the two adjacent coupled-edge signatures; our \(J\) makes that indexing explicit. The sources also use a shifted notation for a middle slice when constructing a single-edge 3-cycle. That auxiliary word is a composition of parallel slice turns in the fixed-center model; it does not add a central generator to the law above.

        3. Why the conditions are sufficient

          The converse uses localized effects rather than a counting guess. First use a face turn, or an internal slice turn, to correct the common parity signs. The sign equations ensure that after this step every permutation factor is even. Commutators and conjugates then supply the required local subgroups:

          • a corner 3-cycle, for example \([[R:U],D]\), generates \(A_8\) on corners;
          • the single-edge 3-cycle \([F,[R:S]]\) generates \(A_{12}\) in the odd case, with the middle-slice symbol interpreted by the parallel-slice composition just described;
          • \([[R_j^{-1},D_i],F^{-1}]\) gives 3-cycles on a center-corner orbit when \(i=j\) and on a center-edge orbit when \(i\ne j\);
          • the analogous coupled-edge commutator gives \(A_{24}\) on each coupled-edge orbit. In even size the source's explicit odd permutation on that orbit upgrades it to \(S_{24}\); in odd size the parity row keeps it at \(A_{24}\).

          Conjugating these words moves each 3-cycle through its whole orbit. Thus the local subgroups clear the corner, single-edge, center-corner, center-edge and coupled-edge permutations in turn. Finally, commutators that twist two corners in opposite directions generate the kernel of the corner-sum map \((\mathbb Z_3)^8\to \mathbb Z_3\); the corresponding edge-flip commutators generate the kernel of the single-edge sum. The coupled-edge orientation vector is already zero by its law. The parity and orientation equations are exactly what is needed for each cleanup step, so reversing the resulting word reaches every tuple satisfying the displayed conditions. This is the constructive sufficiency argument of [salkinder2021nxnxn, Theorems 10--18]; it is a subgroup generation argument, not an assertion that one universal short algorithm solves every size.

          There are two boundaries to keep visible. The theorem concerns labelled positions and a fixed frame; a colour-only observation quotients by equal labels and may identify several legal labelled states. Also, Bonzio's and Salkinder's orbit labels are source conventions, so a numerical state count must be transported through the index sets before comparison. The next three-layer chapter specializes the same action to \(N=3\), where all center and coupled-edge factors disappear and the three familiar constraints can be worked through with explicit face generators.

          References

            Reference. Nxnxn rubik’s cubes and god’s number [salkinder2021nxnxn]

            @misc{salkinder2021nxnxn,
 title = {nxnxn Rubik's Cubes and God's Number},
 author = {Daniel Salkinder},
 year = {2021},
 url = {https://arxiv.org/abs/2112.08602},
 primaryclass = {math.CO},
 archiveprefix = {arXiv},
 eprint = {2112.08602}
}

            Reference. On the nxnxn rubik’s cube [bonzio2018nxnxn]

            • 2018
            • Stefano Bonzio, Andrea Loi, Luisa Peruzzi
            @article{bonzio2018nxnxn,
 title = {On the nxnxn Rubik's Cube},
 author = {Bonzio, Stefano and Loi, Andrea and Peruzzi, Luisa},
 year = {2018},
 journal = {Mathematica Slovaca},
 volume = {68},
 number = {5},
 pages = {957--974},
 publisher = {De Gruyter}
}

            Backlinks

              How the cube models connect [frap-001A]

                An abstract cube turn records incidence positions and restricts a shape symmetry to one facet. A layered-cube turn records cubies, stickers, frames and allowed layers. For the ordinary three-layer cube these are two descriptions of the same action; the map below identifies their coordinates.

                From facet incidence to the 54-sticker state

                  The facet calculation in one ordinary cube face turn selects the facet-colored part of the incidence set. Put the cube's surface cells at \(p\in \{-1,0,1\}^3\ \text {with}\ \lVert p\rVert _\infty =1\). A visible sticker is a pair \((p,n)\), where \(n=\operatorname {sgn}(p_i)e_i\) for each nonzero coordinate \(p_i\). Thus a corner has three visible normals, an edge has two, and a face center has one:

                  \[8\cdot 3+12\cdot 2+6=54.\]

                  The vertex--facet incidences in the facet-colored set map to the pairs above by sending a cube vertex, edge, or facet to its surface-cell coordinate and its incident facet normal. Conversely, the zero-coordinate pattern of \(p\) determines the cubie (three nonzero entries for a corner, two for an edge, one for a centre), and \(n\) determines the incident facet, so this is a bijection between the 54 incidences and the 54 pairs. The cubie counts and this incidence decomposition follow the general model in [bonzio2018nxnxn, Sections 2--3]; the displayed pairing and 54-count are the calculation used here. For \(N=3\), group these pairs into eight corner and twelve edge sticker fibres, leaving the six center positions as fixed frame marks. A legal outer-face move preserves this partition and carries the labels in a corner fibre or edge fibre to the corresponding destination fibre. Write \(C_p=\{(p,\operatorname {sgn}(p_i)e_i):p_i\ne 0\}\) for a corner fibre and \(E_p\) for an edge fibre. For a corner, order its normals as \(L_p=(u_p,f_p,r_p)\), where \(u_p=\operatorname {sgn}(p_3)e_3\), \(f_p=\operatorname {sgn}(p_2)e_2\), and \(r_p=\operatorname {sgn}(p_1)e_1\). For an edge, put first the U/D normal when \(p_3\ne 0\), otherwise the F/B normal, and put the other normal second. A move sends \(C_p\) or \(E_p\) to a destination fibre. Let \(k_p\in \{0,1,2\}\) be the index of the transported U/D normal in \(L_{p'}\). If the source cubie has \(p_3=+1\), set \(x_p=k_p\); if it has \(p_3=-1\), set \(x_p=-k_p\) in \(\mathbb Z_3\). Thus each side turn gives two \(+1\) and two \(-1\) corner twists, while U/D turns give zero, matching the U/D-sticker convention in the complete three-layer cube. For an edge, its distinguished normal agrees or disagrees with the first destination label, giving \(z\in \mathbb Z_2\). This explicit extraction fixes each per-fibre bijection and the destination permutation without asserting an unstated cyclic convention. Therefore it induces exactly the fixed-frame coordinates

                  \[(\sigma ,x;\tau ,z)\in S_8\times (\mathbb Z_3)^8 \times S_{12}\times (\mathbb Z_2)^{12}\]

                  used in the complete three-layer cube. More explicitly, if a move sends a corner fibre \(C_p\) to \(C_{p'}\), set \(\sigma (p)=p'\); the destination edge fibre similarly gives \(\tau \). For the positive-\(z\) turn \(q(x,y,z)=(y,-x,z)\) from one ordinary cube face turn,

                  \[((1,1,1),e_z)\longmapsto ((1,-1,1),e_z),\qquad ((1,1,1),e_y)\longmapsto ((1,-1,1),e_x).\]

                  This exhibits both the destination corner and the transported local frame. The induced map is a change of state description for the legal action: it sends a sticker permutation to its corner and edge destination permutations together with the local frame changes. It does not assert that every permutation of the 54 positions is a legal cube move.

                  This also separates two kinds of forgetting. Passing from the incidence action to the displayed coordinates forgets the individual sticker addresses inside each recorded local frame. Passing from labelled cubies to an ordinary colour observation forgets equal-looking labels. The latter is the observation quotient discussed in which moves survive a quotient, and it is a separate identification of states rather than a change in these coordinates.

                  What changes when layers are added

                    For larger \(N\), the same surface-cell description retains the corner and edge fibers but adds coupled-edge and center orbits. Inner layers move those new fibers, and even and odd sizes have different orbit signatures. The all-\(N\) coordinates in the layered-cube law therefore extend the three-layer state description rather than replace it. Their parity and orientation equations are stated for a labelled fixed-frame model; they are not obtained by treating every center sticker as one of the twelve ordinary edges.

                    The transition is consequently a specialization in one direction: the \(N=3\) outer-layer case of the family of layered cubes gives the ordinary fixed-frame action, while the abstract facet action explains why that outer turn moves precisely the selected surface fibers. Whole-cube rotations, marked centers, central slices and colour-only identifications are additional choices, not silent changes of coordinates.

                    What the identification supports

                      The explicit identification supports this chain of state descriptions:

                      1. incidence, flags and a local facet action explain what a move is;
                      2. the layered model and its first law explain which fibres and constraints appear as \(N\) changes;
                      3. the fixed-frame three-layer coordinates make the three familiar constraints and their constructive sufficiency explicit;
                      4. human methods and computer searches use the same state action while assigning different meanings to preservation, cost and optimality;
                      5. a declared move word can be replayed and checked against these state descriptions.

                      Three linked comparisons use the same distinctions. Label transport and branched unfolding study return maps on simplex labels; compatible quotients test whether an identification of sticker observations inherits a move; and scrambling and spectra study which functions retain memory under a declared move law. These are comparisons of transport, forgetting and randomization, not identifications with the fixed-frame cube action.

                      How the notes are organized [frap-000C]

                        The developed mathematics follows one practical question: what data must a move carry so that we can later compose moves and test reachability? Incidence, symmetry and moves answers it in order. First, polytopes, flags and facet symmetry describe the shape and show their role in triangle and tetrahedron examples. Next, facet moves turn a local facet rotation into an action on incidence positions, with tetrahedron and cube calculations that make the construction concrete. Only then do positions, orientations and global constraints separate a piece's destination from transport in its local frame, while transfer and the effect of dimension connect that transport with a group homomorphism and compute the relevant local abelianizations.

                        The cube examples above connect the general mathematics with solving and play. The later accounts use those concrete states and moves as their starting point:

                        The transition between the abstract facet action and the fixed-frame cube coordinates is recorded in how the cube models connect. Read it before the all-\(N\) law if the distinction between incidence positions, cubie fibres and observations is new.

                        1. General Rubik puzzles: the incidence actions, framed orientations and transfer developed below.
                        2. The layered-cube family: piece types, layer turns and conventions as \(N\) varies.
                        3. The layered-cube law: all-\(N\) state coordinates and a constructive characterization of reachability.
                        4. The complete three-layer cube: explicit outer generators, impossible states, the standard count and a worked cleanup.
                        5. Human methods and computer search: recognizable cases, algebraic effects, usability, move costs and optimality. The complete layer-first method supplies one full human construction, and computer solving and efficiency supplies exact small search, admissible bounds and subgroup reduction.
                        6. From move sequences to play: notation, scrambling, recognition, replay and complete solves, including the worked replay and projected lower bound.

                        Further connections relate the developed orientation and transfer calculations to label transport and topology and scrambling and search functions. The quotient-compatible move account gives the criterion for an identification of observations to inherit a puzzle action. Together these results compare transport, randomization and information loss around the same concrete move.

                        Five encounters with a cube [frap-001C]

                          Put a solved three-layer cube on a table and turn its right face once. The move is familiar, but it raises concrete puzzles: which labelled pieces moved, why can a visually plausible arrangement be impossible, and how can another person undo the turn without watching the first hand? Begin with these experiences, then introduce only the distinctions needed to explain them.

                          One turn is larger than the visible face

                            The shaded strip is only the visible part of the layer: the same stickers cross the front, right, back, top, and bottom faces; viewed from above, the top-face part is the corresponding rightmost strip. A whole-cube rotation would change the viewing frame, whereas this face move keeps the six centres fixed and changes one layer. The layered-cube family makes this distinction for every \(N\); the complete three-layer cube spells it out for the six outer generators.

                            A sticker needs a home and a frame

                              A player sees colours; a proof must also track each sticker's cubie and transported local frame. How the cube models connect follows one sticker through incidence position, cubie fibre, and fixed-frame coordinate. A colour quotient forgets labels, while a larger cube adds positions, so the state space changes with the question.

                              A plausible state can still be impossible

                                Take a solved cube and twist one corner in place. Every sticker still has a cubie-sized home, and the three colours at that corner look locally consistent. But the corner orientations now have sum \(1\) modulo \(3\), whereas every face turn preserves sum \(0\). No sequence of legal turns can produce this state.

                                The complete three-layer cube derives the matching edge and permutation restrictions from the six generators. The calculation is useful at the table: it tells a solver when a scrambled-looking object was assembled illegally, and it tells a proof which state space is worth searching.

                                A human algorithm has to preserve these legal states while it borrows pieces from completed areas.

                                A formula often disturbs pieces that were already placed. The disturbance is acceptable only when the setup is undone and the completed pieces return. A working slot, a trigger, and an inverse setup make that promise visible. The complete layer-first method repeats this pattern through the cross, the first layer, the middle layer, and the last layer. Its completeness depends on the preservation and termination checks, not on the formula's familiarity.

                                Undoing a scramble checks the notation

                                  The worked replay and projected lower bound fixes the viewing convention, records the ten tokens, and applies their inverse. Sticker replay returns to solved. A corner-only projected search finds no solution of depth at most two, so the evidence is the bounded statement

                                  \[3\le d_{\rm HTM}\le 10.\]

                                  The inverse word is an upper-bound witness; the projected search is a lower-bound witness. The computer account explains why these checks do not prove that ten is shortest. A state, a word, and a replay log are enough to test the notation before asking for a general solver.

                                  Changing one ingredient changes the problem

                                    Once the move, its frame, its local repair, and its replay are explicit, three variations separate the effects of enlarging the cube, changing the shape, and forgetting labels.

                                    The layered-cube family develops the first variation; the incidence and facet construction in incidence, symmetry and moves develops the second; and which moves survive a quotient? gives the descent condition for the third. Framed orientations, label transport and topology, and scrambling, spectra and search then study the extra information each variation introduces or forgets. Their hypotheses stay attached to the concrete move that prompted the question.

                                    A face turn can therefore be drawn, named, replayed, and checked before it is generalized. The resulting state maps and invariants support later formal certification or measured comparisons, while the examples here make clear which claim each later programme would have to establish.

                                    Related

                                      The complete three-layer cube [frap-0016]

                                        The all-size law has two reasons to be followed by a separate three-layer account. A \(3\times 3\times 3\) cube has no moving centers or coupled edges, and its familiar three tests can be checked directly on the six outer-face generators. We use a fixed frame and unmarked centers. A labelled state is therefore

                                        \[(\sigma ,x;\tau ,z)\in S_8\times (\mathbb Z_3)^8 \times S_{12}\times (\mathbb Z_2)^{12}.\]

                                        Here \(\sigma \) and \(\tau \) record corner and edge positions. The corner frame uses the U/D sticker to record a twist in \(\mathbb Z_3\); an edge frame uses the U/D or F/B sticker convention to record a flip in \(\mathbb Z_2\). The six centers fix the frame but have no recorded orientation. This is the ordinary cube model, rather than a supercube model with marked centers.

                                        Generators and the three obstructions

                                          Write \(R,L,U,D,F,B\) for clockwise quarter-turns as seen from the corresponding face; inverses and squares are allowed. Each turn is a four-cycle on corners and a four-cycle on edges. Thus it changes both permutation signs, leaving

                                          \[\operatorname {sgn}(\sigma )=\operatorname {sgn}(\tau )\]

                                          unchanged. A turn of U or D changes no corner twist in this convention; a turn of another face adds one and subtracts one at opposite corners. The four affected edge frames change in pairs. Consequently every reachable state satisfies

                                          \[\sum _{i=1}^{8}x_i=0\pmod 3,\qquad \sum _{j=1}^{12}z_j=0\pmod 2.\]

                                          These checks are generator checks, so they also cover inverses, powers and arbitrary words. They give immediate impossible states: one twisted corner has corner sum \(1\); one flipped edge has edge sum \(1\); and a swap of two corners alone has odd corner sign but even edge sign. A state can pass all three tests and still require a construction; necessity is not sufficiency. The coordinate convention and invariant calculation agree with the 3-layer case of [salkinder2021nxnxn, Theorems 8--18].

                                          Localized effects and sufficiency

                                            We now describe a constructive cleanup. Let \([a,b]=aba^{-1}b^{-1}\) and \([a:b]=aba^{-1}\). The word

                                            \[c=[[R:U],D]\]

                                            is a three-cycle on corners and fixes the other corner positions. A direct coordinate replay gives this cycle; conjugating by face turns moves it through the connected corner orbit. Hence these conjugates generate \(A_8\). For edges, the outer-turn word

                                            \[e=R\,U^{-1}\,R\,U\,R\,U\,R\,U^{-1}\,R^{-1}\,U^{-1}\,R^2\]

                                            is a three-cycle on three edges and fixes the corners (the familiar U-permutation). Conjugates generate \(A_{12}\). The displayed words are useful witnesses, not a claim that they are short in every metric.

                                            Given a state satisfying the sign equation, apply one face turn when the common sign is odd. The remaining corner and edge permutations are even. Use the conjugated corner cycles to clear \(\sigma \), then the conjugated edge cycles to clear \(\tau \); each cleanup is chosen in the subgroup that fixes the already-cleared piece type. The sign equation is exactly what makes both cleanups possible.

                                            After positions are restored, the orientation kernels are generated by local commutators. The standard corner trigger \((R^{-1}D^{-1}RD)^2\) restores positions while twisting two corners in opposite directions; setup, trigger, and inverse setup transport this effect to any pair. For edges, the local trigger \(F R U R^{-1}U^{-1}F^{-1}\) is combined with its conjugate and inverse setup to give a pure two-edge flip. A direct sticker replay checks that support and the two flipped signs. Their transported effects generate the kernels

                                            \[\ker ((\mathbb Z_3)^8\to \mathbb Z_3)\quad \text {and}\quad \ker ((\mathbb Z_2)^{12}\to \mathbb Z_2).\]

                                            The two displayed sums are therefore the only orientation obstructions. Reversing the cleanup word reaches every tuple satisfying the three tests. This is a constructive subgroup argument; it does not identify a shortest solution.

                                            Count and a worked reachability check

                                              There are \(8!\) corner permutations, \(12!\) edge permutations, \(3^8\) raw corner orientations and \(2^{12}\) raw edge orientations. The sign equation removes a factor of two, and the two orientation sums remove factors of three and two. Thus the number of labelled reachable states is

                                              \[8!\,12!\,3^7\,2^{10} =43\,252\,003\,274\,489\,856\,000.\]

                                              For a concrete reachability check, start from solved and apply the corner cycle \(c\). Its corner permutation is a 3-cycle, its edge permutation is the identity, and its orientation sums remain zero, so it passes every test. Applying \(c^{-1}\) returns to solved. More generally, the cleanup above factors any passing state into a sign correction, even corner and edge permutations, and orientation-pair operations; this gives an explicit solve word once the required cycles and setup positions have been selected.

                                              Unmarked centers explain why a whole-cube rotation is not counted as a generator and why the center colors fix a frame rather than add twelve more coordinates. The all-\(N\) chapter keeps these choices visible when moving centers and coupled edges return. The next chapters can now discuss human methods and computer search using this completely specified 3-layer action.

                                              The family of layered cubes [frap-0003]

                                                To pass from one cube size to another, we first fix a physical model. Let \(C_N\) be an \(N\times N\times N\) array of unit cubies, with coordinates \((i,j,k)\in {1,...,N}^3\). A visible cubie is a surface cell, so at least one coordinate is \(1\) or \(N\). A cell with three boundary coordinates is a corner; with exactly two it is an edge; with exactly one it is a center. This is a description of positions, before we decide which labels a player can see.

                                                The resulting counts are immediate: there are \(8\) corners, \(12(N-2)\) edge cubies, and \(6(N-2)^2\) center cubies. Their sum is the number of surface cells. The count already explains why changing \(N\) is more than making a larger drawing. New edge and center positions appear, and those positions can fall into different move orbits. Bonzio, Loi and Peruzzi use this cubie decomposition in their general model [bonzio2018nxnxn, Sections 2--3]; Salkinder gives a parallel orbit and configuration analysis [salkinder2021nxnxn, Section 2].

                                                The abstract-to-cube handoff is made explicit in how the cube models connect. In the \(N=3\) case, the 54 facet-colored incidence positions group into corner and edge sticker fibers, with six fixed centers supplying the frame; the resulting action is the fixed-frame coordinate model used later. For larger \(N\), the new center and coupled-edge orbits prevent this from being a single ordinary-cube coordinate space.

                                                Layer turns and the fixed frame

                                                  Choose the six coordinate directions as a fixed frame. For an axis \(a\in {x,y,z}\), a depth \(r\) with \(1\le r\le \lfloor N/2\rfloor \), and either boundary of that axis, turn the layer \(a=r\) or \(a=N+1-r\) by a quarter-turn. The turn rotates the other two coordinates and carries every sticker on a moved cubie with it. For odd \(N\), the central layer is deliberately omitted; this is the usual fixed-center convention. Thus \(N=3\) has the six outer-face turns, while \(N=4\) has the outer and adjacent inner layers.

                                                  This choice distinguishes a layer turn from a whole-cube rotation. Whole-cube rotations change the chosen frame and are not generators of the state group here. They can be added as a separate symmetry action, or used to identify observations later. Keeping them out makes the six face directions, and hence a fixed center color scheme, part of the state description.

                                                  A labelled state assigns a distinct name to every physical cubie and, when needed, to every sticker. A move acts by permuting these labels and their local frames. An ordinary colored observation forgets distinctions between equal-colored stickers; a marked or 'supercube' observation retains them. These are different quotients of the same labelled action. In particular, a statement that two center cubies 'look the same' is an observation claim, not a claim that the labelled pieces are interchangeable by a move.

                                                  The coordinate signatures of the orbits

                                                    The useful invariant is the distance of a coordinate from the nearest boundary. A layer turn permutes two coordinates, possibly changes their signs relative to the cube center, and leaves the multiset of these distances unchanged. It therefore preserves corners, edges and centers. On an edge, the one nonboundary coordinate gives a ring index. On a center, the two interior coordinates give an unordered pair of distances. Equal distances give a center-corner position in a concentric ring; unequal distances give a center-edge position. The central pair in an odd cube is the fixed center.

                                                    Write \(K_e=N/2-1\) for even \(N\) and \(K_o=(N-3)/2\) for odd \(N\). The position orbits under the selected layer turns have the following sizes. For even \(N\), the coupled-edge orbits \(E_1,...,E_{K_e}\) each have \(24\) positions. For odd \(N\), there is a single-edge orbit \(E_s\) of size \(12\), together with coupled-edge orbits \(E_1,...,E_{K_o}\) of size \(24\) each. The term coupled records that the edge is away from the odd cube's middle edge position; it does not by itself choose an orientation convention.

                                                    For even \(N\), each center-corner ring \(C_k\), \(1\le k\le K_e\), has \(24\) positions. The center-edge ring \(D_k\), \(2\le k\le K_e\), has \(48(k-1)\) positions, split into \(2(k-1)\) distinct orbits of \(24\). For odd \(N\), there are \(6\) fixed centers, center-corner rings \(C_k\) of size \(24\), and center-edge rings \(D_k\) of size \(24(2k-1)\), each for \(1\le k\le K_o\). Each odd ring is likewise a union of \(2k-1\) orbits of \(24\). Summing these rows gives \(6(N-2)^2\) centers in both parities. The formulas expose the distinction between a position orbit and an orientation coordinate: a one-sticker center has no orientation unless it is specially marked, while a coupled edge can have a convention-dependent orientation record.

                                                    These sets are not merely a count. Face turns carry a representative edge around the four faces of its ring; an inner layer at depth \(k\), conjugated by outer turns, carries it to the other axis directions. The same construction carries a center-corner representative around its ring and a center-edge representative around every position with the same ordered distance-pair or chirality label. Hence each displayed \(24\)-position set is a move orbit, while the distance signature proves that different rows cannot mix. This is the coordinate part of the model; legality constraints on the permutations are a separate question.

                                                    Small cubes make the parity split visible

                                                      For \(N=2\), \(K_e=0\): there are only the eight corners. For \(N=3\), \(K_o=0\): the pieces are eight corners, twelve single edges and six fixed centers. This is the familiar cube, but the description explains why its centers do not move in the chosen layer family.

                                                      For \(N=4\), \(K_e=1\): there are eight corners, one coupled-edge orbit of \(24\), and one center-corner orbit of \(24\); there are no fixed centers and no center-edge ring. For \(N=5\), \(K_o=1\): there are eight corners, twelve single edges, one coupled-edge orbit of \(24\), six fixed centers, one center-corner orbit of \(24\), and one center-edge orbit of \(24\). The latter has \(8+36+54=98\) visible cubies, the familiar Professor's Cube count. These examples show what changes with parity without claiming that the same labels or orientation coordinates are used by every source.

                                                      The ordinary cube action is the \(N=3\) outer-layer case of this model. The next chapter can therefore state constraints for a fixed three-layer convention, while the all-\(N\) first law must still reconcile the labelled, colored and coupled-edge conventions in Bonzio--Loi--Peruzzi and Salkinder. Salkinder writes all slice rotations, including a central slice in odd size, and then treats its use through a shifted middle-slice convention; this is a genuine convention choice, not a harmless notation change. We use the fixed-center generators above and leave the translation between these move families explicit. A necessary invariant will not be promoted to a sufficiency theorem until that reconciliation is written out.